Finite Group Databasebeta

← #49161925

Subgroups — #49161925

Order 37504.

688 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627594C211yesmaximal, characteristic
2#34036031C211yesmaximal, characteristic, complemented
3#34036030C211yesmaximal, characteristic, complemented
4#34036029C211yesmaximal, characteristic, complemented
5#34036022C211yesmaximal, characteristic, complemented
6#34036021C211yesmaximal, characteristic, complemented
7#34036020C211yesmaximal, characteristic, complemented
8#34036019C211yesmaximal, characteristic, complemented
9#34036018C211yesmaximal, characteristic, complemented
10#34036007C211yesmaximal, characteristic, complemented
11#34036007C211yesmaximal, characteristic, complemented
12#34036005C211yesmaximal, characteristic, complemented
13#34035984C211yesmaximal, characteristic, complemented
14#34035984C211yesmaximal, characteristic, complemented
15#388×C293C211yesmaximal, characteristic, complemented
16C24×D293C2211yescharacteristic
17C22×D1172C2211yescharacteristic
18C22×D1172C2211yescharacteristic
19C22×C22⋊D293C2211yescharacteristic
20C22×C4×D293C2211yescharacteristic, complemented
21#22627594C418yescomplemented
22C2×C4⋊Dic293C2211yescharacteristic, complemented
23C2×#17989781C2211yescharacteristic, complemented
24C2×C22⋊Dic293C2211yescharacteristic
25C2×C22⋊Dic293C2211yescharacteristic
26C2×#17989779C2211yescharacteristic
27C2×#17989779C2211yescharacteristic
28C2×#17989779C2211yescharacteristic
29C2×#17989779C2211yescharacteristic
30C2×#17989779C2211yescharacteristic
31C2×#17989779C2211yescharacteristic
32C2×#17989779C2211yescharacteristic
33C2×#17989779C2211yescharacteristic
34C2×#17989779C2211yescharacteristic
35C2×#17989779C2211yescharacteristic
36C2×#17989779C2211yescharacteristic, complemented
37C2×#17989779C2211yescharacteristic, complemented
38C2×#17989779C2211yescharacteristic, complemented
39C2×C4×Dic293C2211yescharacteristic, complemented
40C2×C4×Dic293C2211yescharacteristic, complemented
41#22627531C2211yescharacteristic, complemented
42#22627531C2211yescharacteristic, complemented
43#22627531C2211yescharacteristic, complemented
44#22627531C2211yescharacteristic, complemented
45C2×C4⋊C4×C293C2211yescharacteristic, complemented
46C2×C22⋊C4×C293C2211yescharacteristic
47C2×C22⋊C4×C293C2211yescharacteristic
48C2×C22⋊C4×C293C2211yescharacteristic
49C2×C42×C293C2211yescharacteristic, complemented
50#95×C293C2211yescharacteristic, complemented
51#95×C293C2211yescharacteristic, complemented
52C23×D293—41no—
53C23×D293—41no—
54C23×D293—41no—
55C23×D293—41no—
56C23×D293—41no—
57C23×D293—41no—
58C23×D293C2×C414yescomplemented
59C23×D293C2311yescharacteristic
60C23×D293C2311yescharacteristic
61C23×D293C2311yescharacteristic
62C2×D1172—41no—
63C2×D1172—41no—
64C2×D1172—41no—
65C2×D1172—41no—
66C2×D1172C2×C414yes—
67C2×D1172C2×C414yes—
68C2×C22⋊D293—41no—
69C2×C22⋊D293—41no—
70C2×C22⋊D293C2×C414yes—
71C22×Dic293C2311yescharacteristic
72C22×Dic293C2311yescharacteristic
73C22×Dic293C2311yescharacteristic
74C22×Dic293C2311yescharacteristic
75C22×Dic293C2311yescharacteristic
76C2×C4×D293—41no—
77C2×C4×D293—41no—
78C2×C4×D293—41no—
79C4⋊Dic293—41no—
80#17989781—41no—
81C22⋊Dic293—41no—
82C22⋊Dic293—41no—
83#17989779—41no—
84#17989779—41no—
85#17989779—41no—
86#17989779—41no—
87#17989779—41no—
88#17989779—41no—
89#17989779—41no—
90#17989779—41no—
91#17989779—41no—
92#17989779—41no—
93#17989779—41no—
94#17989779C2×C414yescomplemented
95#17989779C2×C414yescomplemented
96C4×Dic293—41no—
97C4×Dic293—41no—
98C24×C293C2311yescharacteristic
99C22×C4×C293C2311yescharacteristic
100C22×C4×C293C2311yescharacteristic
101C22×C4×C293C2311yescharacteristic
102C22×C4×C293C2311yescharacteristic
103C22×C4×C293C2311yescharacteristic
104C22×C4×C293C2311yescharacteristic
105C4⋊C4×C293—41no—
106C22⋊C4×C293—41no—
107C22⋊C4×C293—41no—
108C22⋊C4×C293C2×C414yescomplemented
109C42×C293—41no—
110C22×D293—21no—
111C22×D293—21no—
112C22×D293—21no—
113C22×D293—21no—
114C22×D293—21no—
115C22×D293—21no—
116C22×D293—21no—
117C22×D293—21no—
118C22×D293—21no—
119C22×D293—21no—
120C22×D293—21no—
121C22×D293—21no—
122C22×D293—21no—
123C22×D293—21no—
124C22×D293—21no—
125C22×D293—21no—
126C22×D293—21no—
127C22×D293—21no—
128C22×D293—44no—
129C22×D293—44no—
130C22×D293—44no—
131C22×D293—44no—
132C22×D293—44no—
133C22×D293—44no—
134C22×D293—44no—
135C22×D293C22×C412yes—
136C22×D293C22×C412yes—
137C22×D293C22×C412yes—
138D1172—44no—
139D1172—44no—
140C22⋊D293—44no—
141C2×Dic293—21no—
142C2×Dic293—21no—
143C2×Dic293—21no—
144C2×Dic293—21no—
145C2×Dic293—21no—
146C2×Dic293—21no—
147C2×Dic293—21no—
148C2×Dic293—21no—
149C2×Dic293—21no—
150C2×Dic293—21no—
151C2×Dic293—21no—
152C2×Dic293—21no—
153C2×Dic293—21no—
154C2×Dic293—21no—
155C2×Dic293C22×C412yes—
156C4×D293—44no—
157C23×C293—21no—
158C23×C293—21no—
159C23×C293—21no—
160C23×C293—21no—
161C23×C293—21no—
162C23×C293—21no—
163C23×C293C22×C412yes—
164C23×C293C2411yescharacteristic
165C2×C4×C293—21no—
166C2×C4×C293—21no—
167C2×C4×C293—21no—
168C2×C4×C293—21no—
169C2×C4×C293—21no—
170C2×C4×C293—21no—
171C2×C4×C293—21no—
172C2×C4×C293—21no—
173C2×C4×C293—21no—
174C2×C4×C293—21no—
175C2×C4×C293—21no—
176C2×C4×C293—21no—
177C2×C4×C293—21no—
178C2×C4×C293—21no—
179C2×C4×C293—21no—
180C2×C4×C293—21no—
181C2×C4×C293C22×C412yes—
182C2×C4×C293C22×C412yes—
183C22×C293—22no—
184C22×C293—22no—
185C22×C293—22no—
186C22×C293—41no—
187C22×C293—41no—
188C22×C293—41no—
189C22×C293—41no—
190C22×C293C23×C411yescharacteristic
191C22×C293H(5,𝔽2)11yescharacteristic
192C22×C293H(5,𝔽2)11yescharacteristic
193C22×C293H(5,𝔽2)11yescharacteristic
194C22×C293H(5,𝔽2)11yescharacteristic
195C22×C293H(5,𝔽2)11yescharacteristic
196C22×C293H(5,𝔽2)11yescharacteristic
197C2×D293—22no—
198C2×D293—22no—
199C2×D293—22no—
200C2×D293—22no—
201C2×D293—22no—
202C2×D293—22no—
203C2×D293—22no—
204C2×D293—22no—
205C2×D293—22no—
206C2×D293—416no—
207C2×D293—41no—
208C2×D293—41no—
209C2×D293—41no—
210C2×D293—41no—
211C2×D293—41no—
212C2×D293—41no—
213C2×D293—41no—
214C2×D293—41no—
215C2×D293—41no—
216C2×D293—41no—
217C2×D293—41no—
218C2×D293—41no—
219C4×C293—22no—
220C4×C293—22no—
221C4×C293—41no—
222C4×C293—41no—
223C4×C293—41no—
224C4×C293—41no—
225Dic293—22no—
226Dic293—41no—
227Dic293—41no—
228Dic293—41no—
229Dic293—41no—
230C2×C293—42no—
231C2×C293#51811yescharacteristic
232C2×C293#51811yescharacteristic
233C2×C293#51811yescharacteristic
234C2×C293#56011yescharacteristic
235C2×C293#56011yescharacteristic
236C2×C293#56011yescharacteristic
237C2×C293H(3,𝔽4)11yescharacteristic
238D293—42no—
239D293—42no—
240D293—42no—
241C293#237511yescharacteristic, complemented, Sylow
242#2375—2931nomaximal, Sylow
243C2×#120—2931no—
244#390—2931no—
245#390—2931no—
246#390—2931no—
247#388—2931no—
248#388—2931no—
249#388—2931no—
250#388—2931no—
251#388—2931no—
252#388D29311yescharacteristic, complemented
253#386—2931no—
254#386—2931no—
255#386—2931no—
256#379—2931no—
257#379—2931no—
258C25—2931no—
259C22×D4—2931no—
260C22×D4—2931no—
261C22×D4—2931no—
262C23×C4—2931no—
263#120—2938no—
264C2×C4⋊C4—2931no—
265C2×C4⋊C4—2931no—
266C2×C4⋊C4C2×D29311yescharacteristic, complemented
267C2×C22⋊C4—2931no—
268C2×C22⋊C4—2931no—
269C2×C22⋊C4—2931no—
270C2×C22⋊C4—2931no—
271C2×C22⋊C4—2931no—
272C2×C22⋊C4—2931no—
273C2×C22⋊C4—2931no—
274C2×C22⋊C4—2931no—
275C2×C22⋊C4—2931no—
276C2×C22⋊C4—2931no—
277C2×C22⋊C4—2931no—
278C2×C22⋊C4—2931no—
279C2×C22⋊C4—2931no—
280C2×C22⋊C4—2931no—
281C2×C22⋊C4—2931no—
282C2×C22⋊C4C2×D29311yescharacteristic
283C2×C22⋊C4C2×D29311yescharacteristic
284C2×C22⋊C4C2×D29311yescharacteristic
285C2×C42—2931no—
286C2×C42—2931no—
287C2×C42C2×D29311yescharacteristic, complemented
288#95—2931no—
289#95—2931no—
290#95—2931no—
291#95—2931no—
292#95C2×D29311yescharacteristic, complemented
293#95C2×D29311yescharacteristic, complemented
294C24—2931no—
295C24—2931no—
296C24—2931no—
297C24—2934no—
298C24—11721no—
299C24—11721no—
300C24—11721no—
301C24—11721no—
302C24—11721no—
303C24—11721no—
304C24C22×D29311yescharacteristic
305C2×D4—2934no—
306C2×D4—2934no—
307C2×D4—2934no—
308C2×D4—11721no—
309C2×D4—11721no—
310C2×D4—11721no—
311C2×D4—11721no—
312C2×D4—11721no—
313C2×D4—11721no—
314C22×C4—2931no—
315C22×C4—2931no—
316C22×C4—2931no—
317C22×C4—2931no—
318C22×C4—2931no—
319C22×C4—11721no—
320C22×C4—11721no—
321C22×C4—11721no—
322C22×C4C22×D29311yescharacteristic
323C22×C4C22×D29311yescharacteristic
324C22×C4C22×D29311yescharacteristic
325C22×C4C22×D29311yescharacteristic
326C22×C4C22×D29311yescharacteristic
327C22×C4C22×D29311yescharacteristic
328C4⋊C4—41no—
329C4⋊C4—11721no—
330C4⋊C4—11721no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—2934no—
334C22⋊C4—2934no—
335C22⋊C4—11721no—
336C22⋊C4—11721no—
337C22⋊C4—11721no—
338C22⋊C4—11721no—
339C22⋊C4—11721no—
340C22⋊C4—11721no—
341C22⋊C4—11721no—
342C22⋊C4—11721no—
343C22⋊C4—11721no—
344C22⋊C4—11721no—
345C22⋊C4—11721no—
346C22⋊C4—11721no—
347C22⋊C4—11721no—
348C22⋊C4C4×D29314yescomplemented
349C42—41no—
350C42—11721no—
351C42—11721no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—2932no—
359C23—2932no—
360C23—2932no—
361C23—5861no—
362C23—5861no—
363C23—5861no—
364C23—5861no—
365C23—5861no—
366C23—5861no—
367C23—5861no—
368C23—5861no—
369C23—5861no—
370C23—5861no—
371C23—5861no—
372C23—5861no—
373C23—5861no—
374C23—5861no—
375C23—5861no—
376C23—5861no—
377C23—5861no—
378C23—5861no—
379C23—11724no—
380C23—11724no—
381C23—11724no—
382C23—11724no—
383C23—11724no—
384C23—11724no—
385C23—11724no—
386C23C2×C4×D29312yes—
387C23C23×D29311yescharacteristic
388D4—11724no—
389D4—11724no—
390D4—11724no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—2932no—
408C2×C4—5861no—
409C2×C4—5861no—
410C2×C4—5861no—
411C2×C4—5861no—
412C2×C4—5861no—
413C2×C4—5861no—
414C2×C4—5861no—
415C2×C4—5861no—
416C2×C4—5861no—
417C2×C4—5861no—
418C2×C4—5861no—
419C2×C4—5861no—
420C2×C4—5861no—
421C2×C4—5861no—
422C2×C4—11724no—
423C2×C4C2×C4×D29312yes—
424C2×C4C2×C4×D29312yes—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—5862no—
433C22—5862no—
434C22—5862no—
435C22—5862no—
436C22—5862no—
437C22—5862no—
438C22—5862no—
439C22—5862no—
440C22—5862no—
441C22—117216no—
442C22—11721no—
443C22—11721no—
444C22—11721no—
445C22—11721no—
446C22—11721no—
447C22—11721no—
448C22—11721no—
449C22—11721no—
450C22—11721no—
451C22—11721no—
452C22—11721no—
453C22—11721no—
454C22C22×C4×D29311yescharacteristic
455C22#2262766911yescharacteristic
456C22#2262766911yescharacteristic
457C22#2262767011yescharacteristic
458C22#2262767011yescharacteristic
459C22#2262767011yescharacteristic
460C22#2262767011yescharacteristic
461C4—22no—
462C4—22no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—5862no—
468C4—11721no—
469C4—11721no—
470C4—11721no—
471C4—11721no—
472C2—42no—
473C2—11722no—
474C2—11722no—
475C2—11722no—
476C2#3403660111yescharacteristic
477C2#3403660111yescharacteristic
478C2#3403660311yescharacteristic
479C2#3403686711yescharacteristic
480C2#3403686911yescharacteristic
481C2#3403687011yescharacteristic
482C2#3403687111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D293
commutatorC22×C293C23×C4
Fitting#388×C293C2
socle, cosocleC23×C293C24
prime radicalC2×#22627594C2
hypercenter#388D293
hypocenterC293#2375
LCSG ▷ C22×C293 ▷ C293
UCSC1 ◁ C23 ◁ #388

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #34036031C2semidirect
3. #34036030C2semidirect
4. #34036029C2semidirect
5. #34036022C2semidirect
6. #34036021C2semidirect
7. #34036020C2semidirect
8. #34036019C2semidirect
9. #34036018C2semidirect
10. #34036007C2semidirect
11. #34036007C2semidirect
12. #34036005C2semidirect
13. #34035984C2semidirect
14. #34035984C2semidirect
15. #388×C293C2semidirect
20. C22×C4×D293C22semidirect
21. #22627594C4semidirect
22. C2×C4⋊Dic293C22semidirect
23. C2×#17989781C22semidirect
36. C2×#17989779C22semidirect
37. C2×#17989779C22semidirect
38. C2×#17989779C22semidirect
39. C2×C4×Dic293C22semidirect
40. C2×C4×Dic293C22semidirect
41. #22627531C22semidirect
42. #22627531C22semidirect
43. #22627531C22semidirect
44. #22627531C22semidirect
45. C2×C4⋊C4×C293C22semidirect
49. C2×C42×C293C22semidirect
50. #95×C293C22semidirect
51. #95×C293C22semidirect
58. C23×D293C2×C4semidirect
94. #17989779C2×C4semidirect
95. #17989779C2×C4semidirect
108. C22⋊C4×C293C2×C4semidirect
241. C293#2375semidirect
252. #388D293semidirect
266. C2×C4⋊C4C2×D293semidirect
287. C2×C42C2×D293semidirect
292. #95C2×D293semidirect
293. #95C2×D293semidirect
348. C22⋊C4C4×D293semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#226275942
#340360312
#340360302
#340360292
#340360222
#340360212
#340360202
#340360192
#340360182
#340360072
#340360052
#340359842
#388×C2932
#2375293